All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is incorrect to interpret such incidents by unseen matters like jinn and the like. What happened to your parents may be due to normal matters and trials that afflict a person, so first of all, we advise you to be patient with her as there is much good in being patient. For more benefit on the excellence of patience, please refer to Fatwa 83577.
You should supplicate Allaah to cure them [your parents] and they should consult specialist doctors and perform Ruqyah as it is beneficial in all cases, Allaah willing. For more benefit on Ruqyah, please refer to Fatwa 88705.
As regards what is mentioned that your mother should personally go to apologize to the jinn for what happened, then this is not permissible and this is superstition; it is most likely that this has a relation with the belief of some non-Muslim nations, like Hindus and the like. Moreover, if such an apology takes place, then this includes a prohibition which is that the jinni becomes more oppressive against the human being. The Prophet heard a man saying: "May Satan become miserable!" Thereupon, the Prophet said: "Do not say so because if you say so, Satan becomes bigger as big as a mountain and he would say: "It is by my force that I knocked him down.", but if you say “Bismillaah” (In the Name of Allaah), he would become smaller like a fly." [Ahmad] This narration is evidence that Satan becomes bigger and if one apologizes to him, then this makes him bigger and more oppressive.
Therefore, you should recite the Quran at home especially Chapter Al-Baqarah (the Cow) [Quran 2] and seek refuge in Allaah from all evil.
For more benefit, please refer to Fataawa 87433 and 86197.